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Sequences And Series

Question
CBSEENMA11014979
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Find the co-ordinates of a point equidistant from points A (a, 0, 0), B (0, b, 0), C (0, 0, c) and O (0, 0, 0).

Solution

Let P (x, y, z) be the required point and
     O (0, 0, 0), A (a, 0, 0), B (0, b, 0) and C (0, 0, c) be the given points.
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                                         open vertical bar PO close vertical bar space equals space open vertical bar PA close vertical bar
rightwards double arrow            square root of left parenthesis straight x minus 0 right parenthesis squared plus left parenthesis straight y minus 0 right parenthesis squared plus left parenthesis straight z minus 0 right parenthesis squared end root space equals space square root of left parenthesis straight x minus straight a right parenthesis squared plus left parenthesis straight y minus 0 right parenthesis squared plus left parenthesis straight z minus 0 right parenthesis squared end root
rightwards double arrow                                   space space space space straight x squared plus straight y squared plus straight z squared space equals space straight x squared plus straight y squared plus straight z squared minus 2 ax plus straight a squared
rightwards double arrow                                        2 ax space equals space straight a squared space rightwards double arrow space straight x space equals space straight a over 2
                                                  open vertical bar PO close vertical bar space equals space open vertical bar PB close vertical bar
rightwards double arrow             square root of left parenthesis straight x minus 0 right parenthesis squared plus left parenthesis straight y minus 0 right parenthesis squared plus left parenthesis straight z minus 0 right parenthesis squared end root space equals space square root of left parenthesis straight x minus 0 right parenthesis squared plus left parenthesis straight y minus straight b right parenthesis squared plus left parenthesis straight z minus 0 right parenthesis squared end root
rightwards double arrow                              straight x squared plus straight y squared plus straight z squared space equals space straight x squared plus straight y squared plus straight z squared minus 2 by plus straight b squared space rightwards double arrow space 2 by space equals space straight b squared space rightwards double arrow space straight y space equals space straight b over 2
Also,                                           space space space space space space open vertical bar PO close vertical bar space equals space open vertical bar PC close vertical bar
rightwards double arrow          square root of left parenthesis straight x minus 0 right parenthesis squared plus left parenthesis straight y minus 0 right parenthesis squared plus left parenthesis straight z minus 0 right parenthesis squared end root space equals space square root of left parenthesis straight x minus 0 right parenthesis squared plus left parenthesis straight y minus 0 right parenthesis squared plus left parenthesis straight z minus straight c right parenthesis squared end root
rightwards double arrow                                 space space space straight x squared plus straight y squared plus straight z squared space equals space straight x squared plus straight y squared plus straight z squared minus 2 cz plus straight c squared space rightwards double arrow space 2 cz space equals space straight c squared space rightwards double arrow space straight z space equals space straight c over 2
Hence, the required point is straight P space left parenthesis straight x comma space straight y comma space straight z right parenthesis space left right arrow space open parentheses straight a over 2 comma space straight b over 2 comma space straight c over 2 close parentheses

Some More Questions From Sequences and Series Chapter

Find the octant in which the following points lie:
A (-3,1, 2), B (-3, 1, -2), C (-3, -1,-2), D (3, -1, -2) and E (3,1, 2).

A point lies on the x-axis at a distance of 5 units towards OX'. What are its co-ordinates?

A point lies on z-axis at a distance of 3 units in the direction OZ. Write its co-ordinates.

A point lies on y-axis and is at a distance of 4 units in the direction OY'. Write its co-ordinates.

A point lies on XY-plane with distances 4 units along OX and 3 units along OY. Write its co-ordinates.

A point lies on YZ-plane with distances 4 units along OY and 5 units along OZ'. Write its co-ordinates.

A point lies on ZX-plane with distances 3 units along OX' and 5 units along OZ'. Write its co-ordinates.

Find the perpendicular distance of point A (-3, 2, -4) from:
(a) ZX-plane      (b)XY-plane         (c) YZ-plane.

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